Results 1 to 12 of 12

Math Help - Prove that E is disconnected

  1. #1
    Senior Member
    Joined
    Jan 2010
    Posts
    273

    Angry Prove that E is disconnected

    Prove that a closed subset E of a metric space (M,d) is disconnected iff there exists disjoint nonempty closed sets E1, E2 such that E=E1UE2.

    for the --> direction, we have E is disconnected, then by definition there exists nonempty disjoint open sets A,B s.t E=AUB, how do I turn this into a union of 2 closed sets, we can't just say since E is closed, then its subsets is closed, right?

    also need some hint for the other direction.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21

    Re: Prove that E is disconnected

    Quote Originally Posted by wopashui View Post
    Prove that a closed subset E of a metric space (M,d) is disconnected iff there exists disjoint nonempty closed sets E1, E2 such that E=E1UE2.

    for the --> direction, we have E is disconnected, then by definition there exists nonempty disjoint open sets A,B s.t E=AUB, how do I turn this into a union of 2 closed sets, we can't just say since E is closed, then its subsets is closed, right?

    also need some hint for the other direction.
    Note that E-A=B and so E-A is open, so A is closed. Similarly, B is closed. Ta-da!
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member
    Joined
    Jan 2010
    Posts
    273

    Re: Prove that E is disconnected

    Quote Originally Posted by Drexel28 View Post
    Note that E-A=B and so E-A is open, so A is closed. Similarly, B is closed. Ta-da!
    so i need to let E1=A and E2=B, then why is A,B are disconnection of E?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21

    Re: Prove that E is disconnected

    Quote Originally Posted by wopashui View Post
    so i need to let E1=A and E2=B, then why is A,B are disconnection of E?
    I'm sorry, what do you mean there? The idea is that if \{A,B\} form a disconnection of X (they are disjoint, open, non-empty, and union to X) then A,B are both closed, and so can clearly be taken to be your E_1,E_2, no?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Senior Member
    Joined
    Jan 2010
    Posts
    273

    Re: Prove that E is disconnected

    ok, I have done one direction, I need to do the --> direction now, supposing E is disconnected, there exists disjoint E1,E2 subset of E, such that E=E1UE2, and there exist open sets A,B such that E1 subset of A and E2 subset of B.
    We need to show that E1, E2 are closed, since we know E is closed, what can we say about E1 and E2? E=E1UE2 implies E1, E2 are closed?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,966
    Thanks
    1785
    Awards
    1

    Re: Prove that E is disconnected

    Quote Originally Posted by wopashui View Post
    ok, I have done one direction, I need to do the --> direction now, supposing E is disconnected, there exists disjoint E1,E2 subset of E, such that E=E1UE2, and there exist open sets A,B such that E1 subset of A and E2 subset of B.
    We need to show that E1, E2 are closed, since we know E is closed, what can we say about E1 and E2? E=E1UE2 implies E1, E2 are closed?
    It is really difficult to follow where you are from the above.
    Please state in an "if...then..." format which half you need to prove now.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Senior Member
    Joined
    Jan 2010
    Posts
    273

    Re: Prove that E is disconnected

    sorry for the confusion, now i need to prove if a closed subset E of a metric space (M,d) is disconnected , then there exists disjoint nonempty closed sets E1, E2 such that E=E1UE2.


    so supposing E is disconnected, there exists disjoint E1,E2 subset of E, such that E=E1UE2, and there exist open sets A,B such that E1 subset of A and E2 subset of B.
    We need to show that E1, E2 are closed, since we know E is closed, what can we say about E1 and E2? E=E1UE2 implies E1, E2 are closed?
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,966
    Thanks
    1785
    Awards
    1

    Re: Prove that E is disconnected

    Quote Originally Posted by wopashui View Post
    sorry for the confusion, now i need to prove if a closed subset E of a metric space (M,d) is disconnected , then there exists disjoint nonempty closed sets E1, E2 such that E=E1UE2.
    I think I know what definition of connect set you are using.
    If E is not connected the there exists two disjoint open sets A~\&~B each having non-empty intersection with E such that E\subset A\cup B.
    Let E_1=E\cap A~\&~E_2=E\cap B. Clearly E_1\cup E_2=E..
    Now if x is a limit point of E_i then x is a limit poiint of E.
    Therefore, x\in E_i because of disjoint open sets.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Senior Member
    Joined
    Jan 2010
    Posts
    273

    Re: Prove that E is disconnected

    Quote Originally Posted by Plato View Post
    I think I know what definition of connect set you are using.
    If E is not connected the there exists two disjoint open sets A~\&~B each having non-empty intersection with E such that E\subset A\cup B.
    Let E_1=E\cap A~\&~E_2=E\cap B. Clearly E_1\cup E_2=E..
    Now if x is a limit point of E_i then x is a limit poiint of E.
    Therefore, x\in E_i because of disjoint open sets.


    sorry, are you just restating the definition, or this is the proof? I need to show that given E is closed and disconnected, then exists disjoint nonempty closed sets E1, E2 such that E=E1UE2.
    Follow Math Help Forum on Facebook and Google+

  10. #10
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,966
    Thanks
    1785
    Awards
    1

    Re: Prove that E is disconnected

    Quote Originally Posted by wopashui View Post
    I need to show that given E is closed and disconnected, then exists disjoint nonempty closed sets E1, E2 such that E=E1UE2.
    I showed exactly that. E_1~\&~E_2 are disjoint closed non-empty sets the union of which is E.
    Now what do you not understand?
    Follow Math Help Forum on Facebook and Google+

  11. #11
    Senior Member
    Joined
    Jan 2010
    Posts
    273

    Re: Prove that E is disconnected

    Quote Originally Posted by Plato View Post
    I showed exactly that. E_1~\&~E_2 are disjoint closed non-empty sets the union of which is E.
    Now what do you not understand?
    for the last two lines, if the limit point is in the set, the set is closed, but what you are stating is a bit confused me
    Follow Math Help Forum on Facebook and Google+

  12. #12
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,966
    Thanks
    1785
    Awards
    1

    Re: Prove that E is disconnected

    Quote Originally Posted by wopashui View Post
    for the last two lines, if the limit point is in the set, the set is closed, but what you are stating is a bit confused me
    We want to show that E_1 is closed.
    A closed set contains all of its limit points.
    So if t is a limit point of E_1 it is also a limit point of E and because E_1\subset A we know that t cannot be in E_2 so t\in E_1. Thus E_1 contains all of its limit points so it is closed.
    Likewise for E_2.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Disconnected Graphs
    Posted in the Advanced Math Topics Forum
    Replies: 0
    Last Post: March 8th 2012, 07:16 AM
  2. Non-empty, compact, disconnected and limit points
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: March 7th 2012, 07:23 AM
  3. If M is disconnected....
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: April 6th 2010, 01:51 PM
  4. Proof of disconnected sets
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: December 14th 2008, 04:58 PM
  5. [SOLVED] Disconnected Set
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: September 24th 2008, 06:33 PM

Search Tags


/mathhelpforum @mathhelpforum